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The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional ...
To see this, first note that there are 2 n ordered pairs of complementary subsets A and B. In one case, A is empty, and in another B is empty, so 2 n − 2 ordered pairs of subsets remain. Finally, since we want unordered pairs rather than ordered pairs we divide this last number by 2, giving the result above.
The vertices are labeled with ordered pairs (x, y), where x and y are integers between 1 and 9. In this case, two distinct vertices labeled by (x, y) and (x′, y′) are joined by an edge if and only if: x = x′ (same column) or, y = y′ (same row) or,
Given a set X, a relation R over X is a set of ordered pairs of elements from X, formally: R ⊆ { (x,y) | x, y ∈ X}. [2] [10] The statement (x,y) ∈ R reads "x is R-related to y" and is written in infix notation as xRy. [7] [8] The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy.
More generally, a pairing function on a set is a function that maps each pair of elements from into an element of , such that any two pairs of elements of are associated with different elements of , [5] [a] or a bijection from to .
Pairs of spaces occur centrally in relative homology, [1] homology theory and cohomology theory, where chains in are made equivalent to 0, when considered as chains in . Heuristically, one often thinks of a pair ( X , A ) {\displaystyle (X,A)} as being akin to the quotient space X / A {\displaystyle X/A} .